Question

If for a spherical mirror object distance, $u = (50.1 \pm 0.5)cm$ and image distance$v = (20.1 \pm 0.2)cm$, then the focal length of spherical mirror will be
A. $(14.3 \pm 0.1)cm$
B. $(14.3 \pm 0.5)cm$
C. $(30.1 \pm 0.1)cm$
D. $(25.3 \pm 0.5)cm$

Answer 1

Hint Find the focal length using mirror formula using object distance and image distance. Main part is error calculation, for this must know the error calculation in addition, multiplication and division. Use the formula $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$ to find the value of focal length and to find value of error $\Delta f$modify the above formula and use that. Final answer is focal length with error.

Complete step-by-step solution :
Given $u = (50.1 \pm 0.5)cm$ and $v = (20.1 \pm 0.2)cm$.
Putting these values in formula $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$ or $f = \dfrac{{u.v}}{{u + v}}$ we get,
$f = \dfrac{{(50.1) \times (20.1)}}{{(50.1) + (20.1)}} = \dfrac{{1007.01}}{{70.2}} = 14.3cm$
From $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$,
$\dfrac{{\Delta f}}{{{f^2}}} = \dfrac{{\Delta u}}{{{u^2}}} + \dfrac{{\Delta v}}{{{v^2}}}$ or $\Delta f = \Delta u\left( {\dfrac{{{f^2}}}{{{u^2}}}} \right) + \Delta v\left( {\dfrac{{{f^2}}}{{{v^2}}}} \right)$
$\Delta f = 0.5{\left( {\dfrac{{14.3}}{{50.1}}} \right)^2} + 0.2{\left( {\dfrac{{14.3}}{{20.1}}} \right)^2}$
$\Delta f = 0.04 + 0.1 = 0.14cm$
Then $f = 14.3 \pm 0.1$.

Hence the correct answer is option A.

Note:- Error for $\dfrac{{a \pm \Delta a}}{{b \pm \Delta b}}$ id given by $\left( {\dfrac{{\Delta a}}{a} + \dfrac{{\Delta b}}{b}} \right) \times \left( {\dfrac{a}{b}} \right)$ ,error for $(a \pm \Delta a) + (b \pm \Delta b)$ is given by $\pm (\Delta a + \Delta b)$ and for $(a \pm \Delta a) \times (b \pm \Delta b)$ is given by $\pm (\Delta a \times b + \Delta b \times a)$. Focal length is positive then the given mirror is concave mirror because concave mirror has positive focal length and focal length of convex mirror is negative